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<title>Figure 8.15: Linear placement problem</title>
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<h1>Figure 8.15: Linear placement problem</h1>
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<pre class="codeinput">
<span class="comment">% Section 8.7.3, Boyd &amp; Vandenberghe "Convex Optimization"</span>
<span class="comment">% Original by Lieven Vandenberghe</span>
<span class="comment">% Adapted for CVX by Joelle Skaf - 10/24/05</span>
<span class="comment">% (a figure is generated)</span>
<span class="comment">%</span>
<span class="comment">% Placement problem with 6 free points, 8 fixed points and 27 links.</span>
<span class="comment">% The coordinates of the free points minimize the sum of the Euclidean</span>
<span class="comment">% lengths of the links, i.e.</span>
<span class="comment">%           minimize    sum_{i&lt;j) h(||x_i - x_j||)</span>
<span class="comment">% where h(z) = z.</span>

linewidth = 1;      <span class="comment">% in points;  width of dotted lines</span>
markersize = 5;    <span class="comment">% in points;  marker size</span>

<span class="comment">% Input Data</span>
fixed = [ 1   1  -1 -1    1   -1  -0.2  0.1; <span class="comment">% coordinates of fixed points</span>
          1  -1  -1  1 -0.5 -0.2    -1    1]';
M = size(fixed,1);  <span class="comment">% number of fixed points</span>
N = 6;              <span class="comment">% number of free points</span>

<span class="comment">% first N columns of A correspond to free points,</span>
<span class="comment">% last M columns correspond to fixed points</span>

A = [ 1  0  0 -1  0  0    0  0  0  0  0  0  0  0
      1  0 -1  0  0  0    0  0  0  0  0  0  0  0
      1  0  0  0 -1  0    0  0  0  0  0  0  0  0
      1  0  0  0  0  0   -1  0  0  0  0  0  0  0
      1  0  0  0  0  0    0 -1  0  0  0  0  0  0
      1  0  0  0  0  0    0  0  0  0 -1  0  0  0
      1  0  0  0  0  0    0  0  0  0  0  0  0 -1
      0  1 -1  0  0  0    0  0  0  0  0  0  0  0
      0  1  0 -1  0  0    0  0  0  0  0  0  0  0
      0  1  0  0  0 -1    0  0  0  0  0  0  0  0
      0  1  0  0  0  0    0 -1  0  0  0  0  0  0
      0  1  0  0  0  0    0  0 -1  0  0  0  0  0
      0  1  0  0  0  0    0  0  0  0  0  0 -1  0
      0  0  1 -1  0  0    0  0  0  0  0  0  0  0
      0  0  1  0  0  0    0 -1  0  0  0  0  0  0
      0  0  1  0  0  0    0  0  0  0 -1  0  0  0
      0  0  0  1 -1  0    0  0  0  0  0  0  0  0
      0  0  0  1  0  0    0  0 -1  0  0  0  0  0
      0  0  0  1  0  0    0  0  0 -1  0  0  0  0
      0  0  0  1  0  0    0  0  0  0  0 -1  0  0
      0  0  0  1  0 -1    0  0  0  0  0 -1  0  0        <span class="comment">% error in data!!!</span>
      0  0  0  0  1 -1    0  0  0  0  0  0  0  0
      0  0  0  0  1  0   -1  0  0  0  0  0  0  0
      0  0  0  0  1  0    0  0  0 -1  0  0  0  0
      0  0  0  0  1  0    0  0  0  0  0  0  0 -1
      0  0  0  0  0  1    0  0 -1  0  0  0  0  0
      0  0  0  0  0  1    0  0  0  0 -1  0  0  0 ];
nolinks = size(A,1);    <span class="comment">% number of links</span>

fprintf(1,<span class="string">'Computing the optimal locations of the 6 free points...'</span>);

cvx_begin
    variable <span class="string">x(N+M,2)</span>
    minimize ( sum(norms( A*x,2,2 )))
    x(N+[1:M],:) == fixed;
cvx_end

fprintf(1,<span class="string">'Done! \n'</span>);

<span class="comment">% Plots</span>
free_sum = x(1:N,:);
figure(1);
dots = plot(free_sum(:,1), free_sum(:,2), <span class="string">'or'</span>, fixed(:,1), fixed(:,2), <span class="string">'bs'</span>);
set(dots(1),<span class="string">'MarkerFaceColor'</span>,<span class="string">'red'</span>);
hold <span class="string">on</span>
legend(<span class="string">'Free points'</span>,<span class="string">'Fixed points'</span>,<span class="string">'Location'</span>,<span class="string">'Best'</span>);
<span class="keyword">for</span> i=1:nolinks
  ind = find(A(i,:));
  line2 = plot(x(ind,1), x(ind,2), <span class="string">':k'</span>);
  hold <span class="string">on</span>
  set(line2,<span class="string">'LineWidth'</span>,linewidth);
<span class="keyword">end</span>
axis([-1.1 1.1 -1.1 1.1]) ;
axis <span class="string">equal</span>;
title(<span class="string">'Linear placement problem'</span>);
<span class="comment">% print -deps placement-lin.eps</span>

figure(2)
all = [free_sum; fixed];
bins = 0.05:0.1:1.95;
lengths = sqrt(sum((A*all).^2')');
[N2,hist2] = hist(lengths,bins);
bar(hist2,N2);
hold <span class="string">on</span>;
xx = linspace(0,2,1000);  yy = 2*xx;
plot(xx,yy,<span class="string">'--'</span>);
axis([0 2 0 4.5]);
hold <span class="string">on</span>
plot([0 2], [0 0 ], <span class="string">'k-'</span>);
title(<span class="string">'Distribution of the 27 link lengths'</span>);
<span class="comment">% print -deps placement-lin-hist.eps</span>
</pre>
<a id="output"></a>
<pre class="codeoutput">
Computing the optimal locations of the 6 free points... 
Calling Mosek 9.1.9: 81 variables, 39 equality constraints
   For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 39              
  Cones                  : 27              
  Scalar variables       : 81              
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00            
Lin. dep.  - tries                  : 1                 time                   : 0.00            
Lin. dep.  - number                 : 0               
Presolve terminated. Time: 0.00    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 39              
  Cones                  : 27              
  Scalar variables       : 81              
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 12
Optimizer  - Cones                  : 27
Optimizer  - Scalar variables       : 81                conic                  : 81              
Optimizer  - Semi-definite variables: 0                 scalarized             : 0               
Factor     - setup time             : 0.00              dense det. time        : 0.00            
Factor     - ML order time          : 0.00              GP order time          : 0.00            
Factor     - nonzeros before factor : 58                after factor           : 70              
Factor     - dense dim.             : 0                 flops                  : 9.88e+02        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   0.0e+00  2.0e+00  2.8e+01  0.00e+00   0.000000000e+00   -2.700000000e+01  1.0e+00  0.00  
1   7.4e-16  5.4e-01  6.0e+00  1.54e-01   -1.422654910e+01  -2.497615423e+01  2.7e-01  0.01  
2   5.3e-16  1.1e-01  6.4e-01  7.12e-01   -2.001439808e+01  -2.254326248e+01  5.6e-02  0.01  
3   5.6e-16  1.9e-02  4.5e-02  9.26e-01   -2.155200568e+01  -2.198765478e+01  9.4e-03  0.01  
4   1.4e-15  3.4e-03  3.5e-03  9.87e-01   -2.183378074e+01  -2.191307478e+01  1.7e-03  0.01  
5   5.6e-15  1.7e-04  3.9e-05  9.97e-01   -2.190425308e+01  -2.190826316e+01  8.6e-05  0.01  
6   2.2e-14  4.2e-06  1.5e-07  1.00e+00   -2.190816528e+01  -2.190826322e+01  2.1e-06  0.01  
7   1.9e-13  2.9e-07  2.8e-09  1.00e+00   -2.190825700e+01  -2.190826386e+01  1.5e-07  0.01  
8   2.3e-13  3.4e-08  1.1e-10  1.00e+00   -2.190826300e+01  -2.190826379e+01  1.7e-08  0.01  
9   8.2e-13  5.3e-09  6.8e-12  1.00e+00   -2.190826364e+01  -2.190826377e+01  2.7e-09  0.01  
Optimizer terminated. Time: 0.01    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: -2.1908263644e+01   nrm: 1e+00    Viol.  con: 1e-12    var: 0e+00    cones: 0e+00  
  Dual.    obj: -2.1908263768e+01   nrm: 2e+00    Viol.  con: 0e+00    var: 9e-09    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.01    
    Interior-point          - iterations : 9         time: 0.01    
      Basis identification  -                        time: 0.00    
        Primal              - iterations : 0         time: 0.00    
        Dual                - iterations : 0         time: 0.00    
        Clean primal        - iterations : 0         time: 0.00    
        Clean dual          - iterations : 0         time: 0.00    
    Simplex                 -                        time: 0.00    
      Primal simplex        - iterations : 0         time: 0.00    
      Dual simplex          - iterations : 0         time: 0.00    
    Mixed integer           - relaxations: 0         time: 0.00    

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +21.9083
 
Done! 
</pre>
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<img src="placement_lin__01.png" alt=""> <img src="placement_lin__02.png" alt=""> 
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